The next model of common interest is based on the fourth-derivatives
action. If the previous choice, namely, the Einstein-Hilbert action
of GR, is strongly motivated by the success of Einstein's classical
gravitational theory, the fourth-derivative model is motivated by
the consistency conditions of semiclassical gravity. It is known from
the early paper by Utiyama and DeWitt [1]} (see also the
aforementioned books [2]}, [3]}, [4]}, [5]}) that if
the matter fields are quantum, the action of vacuum (i.e., the
gravitational action) of renormalizable theory has to include both
Einstein-Hilbert action (REF ) and the covariant local
fourth-derivative terms
\(S_{HD}\,=\,\int d^4x\sqrt{-g}\,\Big \lbrace a_1 R_{\mu \nu \alpha \beta }^2+ a_2 R_{\mu \nu }^2+ a_3 R^2+ a_4 \operatorname{\square }R\Big \rbrace .\) 
